[1]} show that likelihood ratio tests are most powerful for testing a simple null versus a simple alternative hypotheses. In [2]}, the Karlin-Rubin theorem is viewed as an extension of Neyman-Pearson approach to most powerful testing of composite hypotheses. The Karlin-Rubin theorem applies to the one-dimensional exponential family. With sequential stopping options, some elements of the sample space become impossible, which changes the distributions of statistics. Sufficient statistics become dependent on the random sample size; see [3]}. If statistics belong to the exponential family without early stopping options, then they belong to a curved exponential family when exposed to early stopping options ([4]}, [5]}, [6]}). Section REF  shows that despite being from a curved exponential family, the sequential tests based on likelihood ratios continue to be most powerful for any \(\alpha \) -spending function.
